Answer:
on Quizlet: 1) yes 2) no, complex roots 3) no, irrational roots
Step-by-step explanation:
Rational root theorem is used to determine the possible root of a function.
[tex]\mathbf{(a)\ f(x)= 4x^2 - 25}[/tex]
Set to 0
[tex]\mathbf{4x^2 - 25 = 0}[/tex]
Add 25 to both sides
[tex]\mathbf{4x^2 = 25}[/tex]
Divide both sides by 4
[tex]\mathbf{x^2 = \frac{25}{4}}[/tex]
Take square roots
[tex]\mathbf{x = \pm\frac{5}{2}}[/tex]
The function has rational roots.
This means that: rational root theorem provides a complete list of all roots
[tex]\mathbf{(b)\ g(x)= 4x^2 + 25}[/tex]
Set to 0
[tex]\mathbf{4x^2 + 25 = 0}[/tex]
Add -25 to both sides
[tex]\mathbf{4x^2 = -25}[/tex]
Divide both sides by 4
[tex]\mathbf{x^2 = -\frac{25}{4}}[/tex]
Take square roots
[tex]\mathbf{x = \pm \sqrt{-\frac{25}{4}}}[/tex]
The function has complex roots.
This means that: rational root theorem does not provide a complete list of all roots
[tex]\mathbf{(c)\ h(x)= 3x^2 - 25}[/tex]
Set to 0
[tex]\mathbf{3x^2 - 25 = 0}[/tex]
Add -25 to both sides
[tex]\mathbf{3x^2 = 25}[/tex]
Divide both sides by 4
[tex]\mathbf{x^2 = 8.333}[/tex]
Take square roots
[tex]\mathbf{x = \pm 2.89}[/tex]
The function has irrational roots.
This means that: rational root theorem does not provide a complete list of all roots
Read more about rational roots at:
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